Optimal. Leaf size=30 \[ \frac{2 (a+b x)^2}{5 b c \sqrt{\frac{c}{a+b x}}} \]
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Rubi [A] time = 0.008121, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ \frac{2 (a+b x)^2}{5 b c \sqrt{\frac{c}{a+b x}}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\left (\frac{c}{a+b x}\right )^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (\frac{c}{x}\right )^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int x^{3/2} \, dx,x,a+b x\right )}{b c \sqrt{\frac{c}{a+b x}} \sqrt{a+b x}}\\ &=\frac{2 (a+b x)^2}{5 b c \sqrt{\frac{c}{a+b x}}}\\ \end{align*}
Mathematica [A] time = 0.0134426, size = 21, normalized size = 0.7 \[ \frac{2 c}{5 b \left (\frac{c}{a+b x}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 22, normalized size = 0.7 \begin{align*}{\frac{2\,bx+2\,a}{5\,b} \left ({\frac{c}{bx+a}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38002, size = 23, normalized size = 0.77 \begin{align*} \frac{2 \, c}{5 \, b \left (\frac{c}{b x + a}\right )^{\frac{5}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.22614, size = 99, normalized size = 3.3 \begin{align*} \frac{2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{\frac{c}{b x + a}}}{5 \, b c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.14943, size = 49, normalized size = 1.63 \begin{align*} \begin{cases} \frac{2 a}{5 b c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}} + \frac{2 x}{5 c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}} & \text{for}\: b \neq 0 \\\frac{x}{\left (\frac{c}{a}\right )^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.172, size = 35, normalized size = 1.17 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{2}}{5 \, b c \sqrt{\frac{c}{b x + a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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